Theorem qsqeqor 10902

Description: The squares of two rational numbers are equal iff one number equals the other or its negative. (Contributed by Jim Kingdon, 1-Nov-2024.)

Assertion

Ref	Expression
qsqeqor	⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))

Proof of Theorem qsqeqor

Step	Hyp	Ref	Expression
1		qre 9849	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
2	1	ad3antrrr 492	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 𝐴 ∈ ℝ)
3		simplr 528	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐴)
4		qre 9849	. . . . . . 7 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ)
5	4	ad3antlr 493	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 𝐵 ∈ ℝ)
6		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐵)
7		sq11 10864	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
8	2, 3, 5, 6, 7	syl22anc 1272	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
9		orc 717	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))
10	8, 9	biimtrdi 163	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
11		oveq1 6020	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴↑2) = (𝐵↑2))
12	11	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 = 𝐵 → (𝐴↑2) = (𝐵↑2)))
13		oveq1 6020	. . . . . . . . 9 ⊢ (𝐴 = -𝐵 → (𝐴↑2) = (-𝐵↑2))
14	13	adantl 277	. . . . . . . 8 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (𝐴↑2) = (-𝐵↑2))
15		qcn 9858	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
16		sqneg 10850	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℂ → (-𝐵↑2) = (𝐵↑2))
17	15, 16	syl 14	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → (-𝐵↑2) = (𝐵↑2))
18	17	ad2antlr 489	. . . . . . . 8 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (-𝐵↑2) = (𝐵↑2))
19	14, 18	eqtrd 2262	. . . . . . 7 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2))
20	19	ex 115	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 = -𝐵 → (𝐴↑2) = (𝐵↑2)))
21	12, 20	jaod 722	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
22	21	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
23	10, 22	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
24	17	eqeq2d 2241	. . . . . 6 ⊢ (𝐵 ∈ ℚ → ((𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
25	24	ad3antlr 493	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
26	1	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℝ)
27		simplr 528	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 0 ≤ 𝐴)
28		qnegcl 9860	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
29		qre 9849	. . . . . . . . 9 ⊢ (-𝐵 ∈ ℚ → -𝐵 ∈ ℝ)
30	28, 29	syl 14	. . . . . . . 8 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℝ)
31	30	ad3antlr 493	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
32		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 𝐵 ≤ 0)
33	4	le0neg1d 8687	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
34	33	ad3antlr 493	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
35	32, 34	mpbid 147	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
36		sq11 10864	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 ≤ -𝐵)) → ((𝐴↑2) = (-𝐵↑2) ↔ 𝐴 = -𝐵))
37	26, 27, 31, 35, 36	syl22anc 1272	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) ↔ 𝐴 = -𝐵))
38		olc 716	. . . . . 6 ⊢ (𝐴 = -𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))
39	37, 38	biimtrdi 163	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
40	25, 39	sylbird 170	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
41	21	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
42	40, 41	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
43		0z 9480	. . . . . 6 ⊢ 0 ∈ ℤ
44		zq 9850	. . . . . 6 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
45	43, 44	ax-mp 5	. . . . 5 ⊢ 0 ∈ ℚ
46		qletric 10491	. . . . 5 ⊢ ((0 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
47	45, 46	mpan 424	. . . 4 ⊢ (𝐵 ∈ ℚ → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
48	47	ad2antlr 489	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
49	23, 42, 48	mpjaodan 803	. 2 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
50		qnegcl 9860	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℚ)
51		qre 9849	. . . . . . . . . 10 ⊢ (-𝐴 ∈ ℚ → -𝐴 ∈ ℝ)
52	50, 51	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℝ)
53	52	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → -𝐴 ∈ ℝ)
54		simplr 528	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 𝐴 ≤ 0)
55	1	le0neg1d 8687	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
56	55	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
57	54, 56	mpbid 147	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 0 ≤ -𝐴)
58	4	ad3antlr 493	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 𝐵 ∈ ℝ)
59		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐵)
60		sq11 10864	. . . . . . . 8 ⊢ (((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((-𝐴↑2) = (𝐵↑2) ↔ -𝐴 = 𝐵))
61	53, 57, 58, 59, 60	syl22anc 1272	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) ↔ -𝐴 = 𝐵))
62	61	biimpd 144	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) → -𝐴 = 𝐵))
63		qcn 9858	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
64		sqneg 10850	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2))
65	63, 64	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → (-𝐴↑2) = (𝐴↑2))
66	65	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴↑2) = (𝐴↑2))
67	66	eqeq1d 2238	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((-𝐴↑2) = (𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
68	67	ad2antrr 488	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
69		negcon1 8421	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
70	63, 15, 69	syl2an 289	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
71		eqcom 2231	. . . . . . . 8 ⊢ (-𝐵 = 𝐴 ↔ 𝐴 = -𝐵)
72	70, 71	bitrdi 196	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴 = 𝐵 ↔ 𝐴 = -𝐵))
73	72	ad2antrr 488	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → (-𝐴 = 𝐵 ↔ 𝐴 = -𝐵))
74	62, 68, 73	3imtr3d 202	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → 𝐴 = -𝐵))
75	74, 38	syl6 33	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
76	21	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
77	75, 76	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
78	52	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → -𝐴 ∈ ℝ)
79		simplr 528	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐴 ≤ 0)
80	55	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
81	79, 80	mpbid 147	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐴)
82	30	ad3antlr 493	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
83		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐵 ≤ 0)
84	33	ad3antlr 493	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
85	83, 84	mpbid 147	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
86		sq11 10864	. . . . . . 7 ⊢ (((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 ≤ -𝐵)) → ((-𝐴↑2) = (-𝐵↑2) ↔ -𝐴 = -𝐵))
87	78, 81, 82, 85, 86	syl22anc 1272	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((-𝐴↑2) = (-𝐵↑2) ↔ -𝐴 = -𝐵))
88	65, 17	eqeqan12d 2245	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((-𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
89	88	ad2antrr 488	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((-𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
90	63	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℂ)
91	15	ad3antlr 493	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐵 ∈ ℂ)
92	90, 91	neg11ad 8476	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵))
93	87, 89, 92	3bitr3d 218	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
94	93, 9	biimtrdi 163	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
95	21	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
96	94, 95	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
97	47	ad2antlr 489	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
98	77, 96, 97	mpjaodan 803	. 2 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
99		qletric 10491	. . . 4 ⊢ ((0 ∈ ℚ ∧ 𝐴 ∈ ℚ) → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
100	45, 99	mpan 424	. . 3 ⊢ (𝐴 ∈ ℚ → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
101	100	adantr 276	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
102	49, 98, 101	mpjaodan 803	1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   = wceq 1395   ∈ wcel 2200   class class class wbr 4086  (class class class)co 6013  ℂcc 8020  ℝcr 8021  0cc0 8022   ≤ cle 8205  -cneg 8341  2c2 9184  ℤcz 9469  ℚcq 9843  ↑cexp 10790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-seqfrec 10700  df-exp 10791

This theorem is referenced by:  4sqlem10  12950